232 IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING, VOL. 22, NO. 2, MAY 2009 … · 2017. 4....

232 IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING, VOL. 22, NO. 2, MAY 2009

A General Harmonic Rule Controller for Run-to-RunProcess Control

Fangyi He, Kaibo Wang, and Wei Jiang

Abstract—The existence of initial bias in parameter estimation isan important issue in controlling short-run processes in semicon-ductor manufacturing. Harmonic rule has been widely used in ma-chine setup adjustment problems. This paper generalizes the har-monic rule to a new controller called general harmonic rule (GHR)controller in run-to-run process control. The stability and opti-mality of the GHR controller is discussed for a wide range of sto-chastic disturbances. A numerical study is performed to comparethe sensitivity of the GHR controller, the exponentially weightedmoving average (EWMA) controller and the variable EWMA con-troller. It is shown that the GHR controller is more robust thanthe EWMA controller when the process parameters are estimatedwith uncertainty.

Index Terms—Automatic process control, EWMA, robust con-trol, worst case.

I. INTRODUCTION

M ANY semiconductor manufacturing processes aresuffering from sudden component failures, initial setup

bias, gradual wear of components or aging effects. To produceconforming products, feedback controllers are needed for suchprocess to generate control actions and maintain output ontarget.

The following model has been used extensively in the liter-ature to represent diverse semiconductor processes (see, e.g.,Tsung et al. [23], Tsung and Apley [22], Apley and Kim [1]):

(1)

where denotes the process input recipe at the end of run(beginning of run ) and , which may not be white

noise (WN), denotes the process disturbance that accounts forthe variability in the process. The parameter is called theoffset or intercept and the parameter is called the process gainor slope. It should be noted that the disturbance , modelsdifferent types of process faults illustrated above. Initial bias,

Manuscript received November 09, 2007; revised January 16, 2009. Currentversion published May 06, 2009. The work of K. Wang was supported by theNational Natural Science Foundation of China (NSFC) under Grant 70802034.The work of W. Jiang was supported by The Hong Kong University of Scienceand Technology under Grant HKUST-DAG08/09.EG10.

F. He is with the School of Systems and Enterprises, Stevens Institute of Tech-nology, Hoboken, NJ 07030 USA (e-mail: [email protected]).

K. Wang is with the Department of Industrial Engineering, Tsinghua Univer-sity, Beijing 100084, China (e-mail: [email protected]).

W. Jiang is with the Department of Industrial Engineering and Logistics Man-agement, The Hong Kong University of Science and Technology, Kowloon,Hong Kong (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSM.2009.2017627

sudden process shifts and gradual process drift can all be ex-pressed by this model.

An IMA(1,1) model is a commonly used structure in semi-conductor process control (Box et al. [2], Box and Kramer [4],Janakiram and Keats [12], Montgomery et al. [14], Tsung et al.[23], Tsung and Apley [22], Vander Wiel et al. [26], Vander Wiel[25], Del Castillo and Hurwitz [7], Box and Luceño [3], Luceño[13], Chen and Guo [5], Apley and Kim [1]). An IMA(1,1)disturbance model can characterize the behavior of a non-sta-tionary process, which is

(2)

where is an independently identically distributed (i.i.d.)sequence of random noise with mean 0 and variance , i.e.,

. is the IMA parameter. In order to verifywhether a noise sequence follows an IMA(1,1) model, onemay take one-step lagged difference of the sequence, and fit anMA(1) model to the differences.

Among others, the exponentially weighted moving average(EWMA) controller has been widely used to compensateprocess deviations or faults. To adjust process output, , toits target value , the EWMA controller suggests

Sachs et al. [24] assume that an estimate of the process gainis available prior to the beginning of the control session. How-

ever, to account for the process disturbance , the interceptis estimated recursively using an EWMA equation. In this con-troller, the estimate of at run is denoted by . This estimateis then updated using the following EWMA equation,

where is a parameter that gives more weight tothe most recently observed forecast error of the quality charac-teristic the closer it is to 1. The control law follows

If the process gain is estimated accurately, i.e., , andthe process output at time 0 is on target, i.e., ,the EWMA controller is the minimum MSE (MMSE) controllerwhen the EWMA parameter is set equal to (Box et al.[2], Sachs et al. [24]).

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HE et al.: A GENERAL HARMONIC RULE CONTROLLER FOR RUN-TO-RUN PROCESS CONTROL 233

However, two critical constrains are confronting the semicon-ductor manufacturing practice. First, estimates of process pa-rameter are never accurate; parameter estimation uncertaintiesalways exist. Therefore, controllers have to be robust to inaccu-rate parameter values and guarantee process stability and controlperformance in all circumstances. Second, short-run processesleave narrow space for process adjustment and call for quick ac-tions to improve transient performance. As a result, when initialsetup bias or fault exists, controllers that respond slowly to suchshifts or faults are not capable in new manufacturing scenarios.

Patel and Jenkins [16] presented a scheme to change theEWMA weight adaptively to compensate step and drift dis-turbances. Chen et al. [6] demonstrated that an ARI(3,1)model is more suitable for the metal sputter deposition processunder investigation. To handle this special type of disturbance,the authors developed an extended Kalman filter controller.Wu et al. [28] investigated the effect of metrology delay oncontrol performance. The authors also suggested alternativesto compensate the adverse effect caused by metrology delaywhen the underlying process exhibits nonstationary or highlyautoregressive disturbance. Nonetheless, none of these worksfocuses on compensation of initial bias and short-run processes.

Short-run processes are becoming more and more prevalentdue to high-degree customization, frequent process mainte-nance and the mix-product trend in semiconductor manufac-turing (see, e.g., Sullo and Vandeven [18], Pan [15], Tsiamyrtzisand Hawkins [21]). For example, in a wafer preparing process,all facilities have to be adjusted everyday or even several timesper day to meet technical specifications of different orders.An etching process has to be re-configured frequently when anew batch of wafers with varying critical dimensions arrives.In recent years, production lines designed with fast-switchcapability are frequently seen. When processes are run withsmall batches, the short-term performance of a controller willbecome a critical concern.

However, the EWMA controller’s short-term performancemay be deteriorated due to the following reasons. First, theprocess may have large initial setup bias, i.e., ;Second, the effect of inaccurately estimated process gain pa-rameter, i.e., , is extrusive during the initial stage; Third,the disturbance sequence may be wrongly identified. When

is not an IMA(1,1) process, and even when follows anIMA(1,1) process but the EWMA parameter is not set to

because of inaccurate estimate of , the performance ofthe EWMA controller becomes unpredictable.

Tseng et al. [20] proposed a variable EWMA (VEWMA) con-troller, which allows to vary at each step. That is, is updatedas follows:

where , is a discount factor. The authors show thatthe VEWMA can compensate initial bias faster than the EWMAcontroller. However, in order to setup the optimal VEWMA con-troller, the amount of initial bias has to be known in advanced.The performance of the VEWMA controller may deteriorate ifthis parameter is not estimated accurately.

In this paper, we propose a new control scheme called thegeneral harmonic rule (GHR) controller which has higherrobustness when parameter estimation uncertainties exist andbetter short-term performance than EWMA controller when theprocess output has a large initial bias. We show that the GHRcontroller is optimal when is white noise and . Incomparison with the EWMA controller’s optimality, the GHRcontroller’s optimality is derived without any assumption onthe process initial condition, i.e., the initial bias is taken as anunknown value instead of 0. We also investigate the sensitivityof both GHR and EWMA controllers and show that the GHRcontroller is more robust than the EWMA controller, especiallywhen is overestimated. The sensitivity analysis also extendsto imperfect estimate of the process gain , i.e., when isoverestimated offline ( and ), the GHRcontroller has much better performance than that of EWMAcontroller. This result is very significant since in practice peopletend to use an overestimated rather than an underestimatedone in the EWMA controller due to its stability condition, i.e.,

(Ingolfsson and Sachs [11], Sachs et al. [24]).The stability condition guarantees the control system awayfrom bursting if is not underestimated too much.

The rest of this paper is organized as follows. Section IIpresents the motivation and problem descriptions. Section IIIdiscusses the stability and optimality of the GHR controller.Section IV provides a sensitivity analysis of both GHR con-troller and EWMA controller. A numerical example is presentedin Sections V and VI concludes the paper.

II. THE GHR CONTROLLER

The setup adjustment problem was first studied by Grubbs[9]. Suppose the measurements represent some quality char-acteristic of the items as they are produced at discrete points intime . Grubbs [9] proposed a method for the adjust-ment of the machine in order to bring the process back to targetif at start-up it was off-target by units, where is an unknownvalue. In this section, we will consider the process model (1)used in run-to-run control. Since both and are unknown andneed to be estimated from the offline procedure, the initial biasof the process output is an unknown value. Using the similarderivation of Grubbs’, we shall in the following derive the op-timal controller based on unknown initial bias.

Assume that the disturbance in model (1) follows a familyof processes which could be represented by

(3)

where for and for all . Itincludes many popular random processes. For example, is awhite noise if for , an IMA(1,1) with parameter

if for , an ARMA(1,1) with parameters ( ,) if for , and an ARIMA(1,1,1) with

parameters ( , ) if for .At any time , denote the process mean conditional on the past

outputs as , i.e., , where stands for theobservations . Set to when .Let represent the deviation of from the target value . We



call the initial bias since it comes from the inaccurate estimateof the process parameters at time 0. can be represented as

(4)

where . Then the true value for the first outputis . Note that is known and is observable,so we can measure directly. However, is unknownand cannot be determined since and are unkown. In thefollowing derivation, without loss of generality, assumeso that represents the output’s deviation from target at time .

After producing the first item and observing the first devia-tion , we can adjust the last period’s input by beforemaking the second item. That is,

(5)

From (1), (3), (4) and (5), we know that, after the first adjust-ment,

It follows that .Similarly, an adjustment can be made on the last period’s input,i.e.,

and

(6)

Continuing this iteration, by making the corrections ,etc., in general, the process mean at time condi-

tional on the past observable deviations follows:

(7)

where we specify that and . Our aimis to determine the adjustment coefficients

, which solve the following optimization problem:

(8)

Theorem 1 gives a recursive form to determine givenfor any time .

Theorem 1: If solve the problem (8), then

(9)where

(10)

(11)

The proof of Theorem 1 can be found in Appendix A. Note thatit is obvious from (10) that . Theorem 1 gives us away to obtain the adjustment at any time based on all theadjustments before time .

According to Theorem 1, we can design a new controller asfollows. Suppose the process to be controlled can be modeledas (1) with the disturbance (3), the new control rule is

(12)

where

(13)

and

(14)

At time 0, and , where and are the of-fline estimates of the process parameters and , respectively.It can be seen from (13) that when all and

, reduces to , which is exactly the harmonic rulein the process setup adjustment problem derived by Grubbs [9].We call the new controller General Harmonic Rule (GHR) con-troller and are called the GHR parameters. In partic-ular, for the IMA(1,1) disturbance with parameter ,for all .

III. STABILITY AND OPTIMALITY

For any control scheme to be practical, a fundamental re-quirement is that the process should achieve long-term stability.Although this paper focuses on the performance of short-runmanufacturing processes, it is worthwhile to investigate stabilityconditions for the GHR controller. A process is said to beasymptotically stable if

(15)



The stability of a process ensures that the mean of the processoutput converges to the desired target, while its asymptotic vari-ance remains bounded. The following theorem gives the sta-bility condition for the GHR controller when the disturbancefollows IMA(1,1) process.

Theorem 2: The GHR controller defined in (12)–(14) isasymptotically stable if when the disturbanceis an IMA(1,1) process.

The proof of Theorem 2 can be found in Appendix B. Thiscondition implies that if is of the same sign and larger (inabsolute value) than , then stability is guaranteed. If is ofthe same sign and smaller (in absolute value) than , it mustbe larger (in absolute value) than for stability. If andhave different signs, the process will always be unstable. Thefollowing theorem gives the conditions under which the GHRcontroller is optimal.

Theorem 3: If and is white noise, i.e., forall , the GHR controller is optimal.

The proof of Theorem 3 can be found in Appendix C. Ourproof is very similar to that of Grubbs [9], but from a controller’saspect. In comparison with the EWMA controller’s optimality,the GHR controller is derived from no assumption of . Thatis, if the process disturbance is white noise and , the GHRcontroller is the optimal controller no matter what the processinitial status is.

IV. SENSITIVITY ANALYSIS

In this section, we take the Deep Reactive Ion Etching (DRIE)process in semiconductor manufacturing as an example to studythe performance of the newly proposed GHR controller. TheDRIE process is an important step for forming desired patternson wafers in micro/nano-scale fabrication; it involves complexchemical-mechanical reactions. Wafers to be etched are loadedinto a chamber. The system first releases etching plasma into thechamber to generate trenches subject to designed mask patterns;then in the deposition step, different gases are introduced intothe chamber to generate a protective film on the sidewalls. Theetching and deposition steps repeat alternately until the presetprocessing time is reached or the end-point detection moduleconfirms the correct etching depth. A more detailed illustrationof the etching process is referred to Wang and Tsung [27]. DRIEhas been successfully used in producing photonic crystals, mag-netic nanostructures and MEMS resonators (STS [17]).

One of the key quality characteristics produced by the DRIEprocess is the etched profile. As is shown in Fig. 1. An idealprofile should be vertical, having an angle of 90 against thehorizontal line . The etch/deposition time ratio is usu-ally adjusted to compensate over-etched or under-etched wafersto generate vertical sidewalls. In an analysis presented by Wangand Tsung [27], the authors studied the DRIE process and sug-gested that the slope of the produced profiles can be modeledby:

where is an IMA(1,1) time series with . The param-eters are estimated from a designed experiment.

Fig. 1. Illustration of an etched profile.

As each production run is very time-consuming and it is notpractical to study the performance of the proposed controllerby adjusting the real process, we treat the above Equation asthe true process model and study the performance of the GHRcontroller. The EWMA and VEWMA controllers are also set upfor the same simulated process for comparison. Our comparisonis divided into the following two parts, when is known andunknown.

A. is Known

In many setup adjustment literature, the process gain is usu-ally assumed known and set to 1 (Grubbs [9], Triestsch [19]and Del Castillo et al. [8]). In this section, we first assumeis known or could be accurately estimated offline, i.e., ,and investigate the performance of both GHR and EWMA con-trollers given different offline estimate of (i.e., ). We willfirst assume the disturbance follows an IMA model and thentakes a more general ARIMA(1,1,1) model.

1) is an IMA(1,1) Process: The IMA(1,1)model has beenwidely adopted to characterize disturbances in diverse applica-tions. The conventional EWMA controller has been proved tobe optimal to compensate such a disturbance series (Box et al.[2]). However, certain conditions must be satisfied in order forthe EWMA controller to achieve its optimal performance. First,the process parameters and are both known or accuratelyestimated offline, i.e., and ; Second, the EWMAparameter should be set to which implies that the IMAparameter should be known or accurately estimated offline.In short-run production scenarios, however, it is often impos-sible to obtain accurate estimate of these parameters. In the fol-lowing simulations, we take , ,and as the true parameters of the DRIE process. TheEWMA, VEWMA and GHR controllers are then set up to ad-just the process under different hypothetical settings.

Three cases are studied in the following. In case 1, we assumeis accurately estimated (i.e., ); in case 2, we assume is

overestimated (i.e., ); and in case 3, we assume is under-estimated (i.e., ). In each case, the EWMA parameterand the GHR parameters are all set to . For theVEWMA controller, the initial value of the smoothing param-eter, , is set to . The choice of the discount factor in theVEWMA controller depends on the initial process bias, whichis unknown in real scenarios. Therefore, we choose a moderate



Fig. 2. � is IMA(1,1) process and � is known �� .

Fig. 3. AMSE of the EWMA and GHR Controllers when � is Over- and Under-estimated.

TABLE ITHE AMSE WHEN � IS IMA(1,1)

value, , in the following simulations. The estimated in-tercept, , is varying within a neighborhood of .

We did 10 000 simulations for each case. In each simulation,we run 80 steps for . The MSE of is computed.The Average MSE (AMSE) of the 10 000 simulations is reportedin Tables I–III. The simulation results are graphically shownin Figs. 2 and 3(a) and 3(b). The standard error in the AMSE(SEAMSE) is reported as well, which is defined as

TABLE IITHE AMSE WHEN � IS IMA(1,1)

TABLE IIITHE AMSE WHEN � IS IMA(1,1)

where SDMSE means the standard deviation of mean squareerrors, and the number of replicates used in the simulation is



10 000. Essentially, SEAMSE measures the significance of thedifference of AMSE values among all controllers.

Both Table I and Fig. 2 show the results when . It is seenthat although the EWMA controller is an MMSE controller when

, which achieves the lowest MSE, the GHR controller ismore robust than the EWMA controller when is deviated from

. The VEWMA controller performs in between; it is superior tothe GHR controller when is close to , while is inferior to theGHR controller when deviates far away from . The AMSEof the GHR controller increases much slower than the EWMAcontroller does when deviates from its true value .

Similar patterns are observed when is either overestimatedor underestimated. When is overestimated, as in case2, Fig. 3(a) and Table II show that the AMSE of using theGHR controller is almost consistently smaller than that usingthe EWMA controller for any values and the margin couldbe as big as 13.3% when . The VEMWA controllershows its capability in compensating initial bias, as Tseng etal. [20] demonstrated. When is over estimated, the VEWMAcontroller is slightly better than the GHR controller or equallygood when is close to , while it becomes less favored whenthe difference between and becomes large.

This result has very important implications in practice. AsHunter [10] suggested, the EWMA parameter is usually set toa value between 0.1 and 0.3. This fact implies that is ordinarilyassumed to be between 0.7 and 0.9, although could be smallerthan 0.7 in many cases. When is overestimated using the ruleof thumb, the advantage of the GHR controller over the EWMAcontroller becomes more significant.

Table III and Fig. 3(b) show the results when is underesti-mated . Again, the robustness of the GHR controlleris proved by its slow increasing trend when deviates from

. Now, the VEWMA controller is slightly inferior to the GHRcontroller for all tested values.

2) is an ARIMA(1,1,1) Process: Even though the processwe investigate has an IMA(1,1) distrubance series, it is mean-ingful to study the performance of the GHR controller underother types of disturbance series. In the following simula-tion, we assume that the disturbance follows an generalARIMA(1,1,1) process with parameters and , while itis misidentified as IMA(1,1) process with parameter . Ap-pendix D shows that the estimate of IMA parameter wouldbe around . We consider two cases in the fol-lowing simulations. In case 1, is set to 0.2 and is set to 0.6; Incase 2, is set to 0.6 and is set to 0.2. If these ARIMA(1,1,1)processes are misidentified as IMA(1,1) processes, is around0.4 in case 1 and around 0.5 in case 2. Similar to the aboveexperiment, we replicated 10,000 simulations for each case.In each simulation, we run 80 steps for and computed theAMSE of the 10,000 simulations as shown in Tables IV and V.

Fig. 4 further plots the AMSE versus for case 1 and case2 respectively. Once again, the GHR controller has more robustperformance in both cases. The inaccurate offline estimate ofhas less impact on the GHR controller than on the EWMA con-troller. The VEWMA controller and the GHR controller showssimilar performance. The VEWMA controller’s AMSE is gen-erally larger when is underestimated, while smaller when isoverestimated.

TABLE IVTHE AMSE WHEN � IS ARIMA(1,1,1)

TABLE VTHE AMSE WHEN � IS ARIMA(1,1,1)

B. is Unknown

In most run-to-run production processes, offline sample isusually not large enough to guarantee accuracy of the parameterestimates, especially the process gain . When is unknownand has to be estimated by offline experiments, it is importantto investigate the performance of the GHR controller when allparameters are estimated with uncertainties.

In the following study, the process is still assumed to fol-lowing (1) with being an IMA(1,1) series. True process pa-rameters are chosen as , , and

. Similar to the case when is known, we analyzethree cases here: is known, is overestimated, and is un-derestimated. In each case, the EWMA parameter and theGHR parameters are all set to . The discountfactor of the VEWMA controller is still set to 0.5. Estimatedparameters, and , are allowed to vary with their respectiveneighborhood areas. The simulation results are reported in Ta-bles VI–VIII. Fig. 5 shows the contour plots of AMSE for theGHR and EWMA controller respectively when is known. It iseasy to see that, for both controllers, the AMSE changes slowlyalong the line and achieves the min-imum when and . Comparing with the GHR con-troller, the contour curve of the EWMA controller is more dense,which indicates that the AMSE value increases more quicklywhen deviates from due to poor offlineestimates of and . That is, the GHR controller is more robustthan the EWMA controller when offline parameter estimatesare not accurate, especially when is bigger than . TheVEWMA controller outperforms the GHR controller only whenthe estimated values are close to their respective true values.This tends to be an advantage of the GHR controller whenis overestimated in practice as we discussed before for the con-troller’s stability.

Since using small values of in the EWMA controller is arule of thumb in practice (Hunter [10]), Figs. 6 and 7 furtherpresent contour plots of the AMSE for the three controllerswhen is overestimated and underestimated. Although all



Fig. 4. AMSE of the EWMA and GHR Controllers when� is an ARIMA(1,1,1) process but is misidentified as an IMA(1,1) process. (a) ARIMA with � � ��and � � ��. (b) ARIMA with � � �� and � � ��.

TABLE VITHE AMSE WHEN � IS KNOWN ��

controllers’ performance are worse than that in Fig. 5, theEWMA controller’s performance deteriorates more signifi-cantly; the VEWMA controller is also inferior to the GHR

TABLE VIITHE AMSE WHEN � IS OVERESTIMATED ��

controller for most tested cases when is underestimated. TheGHR controller is seen to be the most robust against parameterdeviations, especially when is greater than .



Fig. 5. Contour plot of AMSE when � is IMA(1,1) process and � is known.

TABLE VIIITHE AMSE WHEN � IS UNDERESTIMATED ��

V. AN ILLUSTRATIVE EXAMPLE WITH INITIAL BIAS

The above section has studied the performance and robust-ness of the GHR controller. In this section, we use a specificparameter setting and study the impact of initial estimation biason the newly proposed controller.

The process is again assumed to follow (1) and the distur-bance is an IMA(1,1) time series. The true parameter set-

tings are , and , while the estimatedvalues are assumed to be , and ,respectively. That is, initial bias exist in estimating all the pa-rameters in the model. The target value .

Fig. 8 shows the process output against time when thethree controllers are applied. The curves show that, the EWMAcontroller requires a moderately large number of runs to bringthe process output to the target when the process has large ini-tial bias . The VEWMA controlleris more efficient than the EWMA controller in removing initialbias. However, the GHR controller is the fastest in bringing theprocess output back to target. After the first 30 runs, the processoutput are almost the same using both controllers. Althoughthis is just a single realization of the control process, it demon-strates the effectiveness of the GHR controller when the initialbias is significant.

To better reveal the difference between the VEWMA andGHR controllers, we further investigated the mean response ofa process with and . We consider differentcases when estimated and , denoted by and , are biased.Fig. 9 shows the mean responses of a process controlled by theVEWMA and GHR controllers under different initial bias. Allthe cases have an equal initial bias . It is seen from (a) thatwhen only is biased, GHR can bring the output back to targetimmediately, while VEWMA consumes several more steps toapproach the target. When is also biased, in case (b), GHRand VEWMA performs closely, while in case (c), GHR outper-forms VEWMA in compensating the initial bias.

To summarize, when the disturbance sequence follows anIMA model and parameters are accurately estimated, theEWMA controller is always suggested. This is supported byboth theory and simulation results. While when the distur-bance model is not IMA or parameters cannot be estimatedaccurately, either VEWMA or GHR is suggested. More specif-ically, as the GHR controller is more robust in most cases,GHR is more favored when initial bias or estimation uncer-tainties is large.

VI. CONCLUSION

In short-run production processes, the performance of anEWMA controller critically depends on offline estimates of



Fig. 6. Contour plot of AMSE when � is IMA(1,1) process and � is overestimated.

Fig. 7. Contour plot of AMSE when � is IMA(1,1) process and � is underestimated.

Fig. 8. The output � against time �.

process parameters. This paper proposes a new controllerbased on the harmonic rule used in machine setup adjustmentproblems. The sensitivity of the new controller is comparedwith the EWMA and VEWMA controllers under differentscenarios of the disturbance parameter estimate, the processoffset and gains estimate, as well as the misidentification of thedisturbance. The short-run performance of the GHR controlleris shown better than that of the EWMA controller when theoffline estimates of the process or disturbance parameters are

inaccurate. Even though the VEWMA can compensate initialbias faster than the EWMA controller, it is inferior to the GHRcontroller when estimated parameter values deviate far fromtheir true settings, especially when is underestimated. Thestability and optimality conditions are also derived for the newcontroller. The sensitivity analysis indicates that the new GHRcontroller is more robust than the EWMA controller undermodel misspecification and parameter estimation uncertainties.

It should be noted that dislike the EWMA controller, whichis optimal for IMA(1,1) disturbance series only, the GHR con-troller can be applied to processes with any general disturbancemodels that follow (3).

Initial bias in parameter estimation can seriously deterioratecontrol performance, especially in short-run processes. TheGHR controller assumes the initial bias is an unknown butfixed value in this research. In many practices, due to frequentprocess setup, initial bias may be better modelled as a randomvariable rather than a fixed value. The GHR controller can beextended to take random initial bias into considerations, whichshould be a topic for further research. This work focuses onsingle-input-single-output processes only. In many semicon-ductor manufacturing scenarios, a process may have multiplecorrelated inputs and outputs. Extending the GHR controllerto multivariate processes is another important topic for futureresearch



Fig. 9. Mean responses of a process controlled by GHR and VEWMA controllers when initial bias exists.

APPENDIX A

Since

the problem (8) is equivalent to

(16)

From (7), we get that

(17)

and

(18)

Denote as the Lagrange multiplier, we set

and equate to zero. Since

(19)

we find that

(20)

That is,

(21)

where and we used .From the side condition and (17), (19), we know

that

(22)

So from (21) and (22), we get (23), found at the bottom of thepage. Since (21) is right for any , let us set

. Then we can get that

(24)

From (23) and (24), we get

(25)

(23)



where

(26)

It is not difficult to get the recursive version of that

APPENDIX B

We only prove the case that and are positive. Note thatfor all when is IMA(1,1) process.

According to the GHR controller’s definition, it’s easy to getfrom (7). So is asymp-

totically stable if and only if there exists a such thatfor all . We only need to prove if ,

then

is satisfied whenever

(27)

is satisfied.Conditional on , from (13) and (27) we can get

(28)

Because of (27), it’s easy to get

(29)

Then from (14) and (29), we know

(30)

So under the conditions (28) and (30), we know

(31)

From (13), (31) and , we can get

that is, . So .

Since we have proved that for all underthe condition , we can find a real number such that

. Then

(32)

So .

APPENDIX C

When and , the GHR controller willdegenerate into the form that

(33)

In order to prove the GHR controller be optimal, we need toprove the following fact: if

are the solutions of the problem (8), thenare the solutions of the problem

(34)

where

From (18) and , we can derive that

When ,



Put into the equation above, we get

(35)

From (7), it’s easy to derive that

When , we can derive from the above equationthat

(36)

So in order to minimize , we need to let

(37)

Then put (35) into (37), we get

thus we finish our proof.

APPENDIX D

Suppose is ARIMA(1,1,1) process with parameter and, i.e.,

(38)

Then

(39)

If we have infinite sample size, i.e., , then

(40)

Now, is misidentified as IMA(1,1) process with parameter .Using moment estimation, we make

(41)

That is,

(42)

ACKNOWLEDGMENT

The authors are grateful to the three referees and the Asso-ciate Editor for many helpful suggestions that have significantlyimproved the quality of this paper.

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Fangyi He received the B.S. and M.S. degrees in sta-tistics from University of Science and Technologyof China (USTC), and the Ph.D. degree in industrialengineering from the School of Systems and Enter-prises, Stevens Institute of Technology, Hoboken, NJ.

After his graduation, he joined the group of Equityand Derivative of BNP Paribas, New York, workingas a quantitative researcher. His research interests in-clude statistical process control, engineering processcontrol, time series analysis, applied probability andstatistical inference.

Dr. He is a member of American Statistical Association (ASA).

Kaibo Wang received the B. Eng. and M. Eng.degrees in mechatronics from Xi’an JiaotongUniversity, Xi’an, China, and the Ph.D. degree inindustrial engineering and engineering managementfrom Hong Kong University of Science and Tech-nology (HKUST), Hong Kong.

He is currently an Assistant Professor in theDepartment of Industrial Engineering at TsinghuaUniversity. He is on the Editorial Board for Journalof the Chinese Institute of Industrial Engineers(JCIIE). He has published papers on Journal of

Quality Technology, Quality and Reliability Engineering International, Inter-national Journal of Production Research and others. His research interestsinclude quality management, statistical process control, and run-to-run processcontrol.

Wei Jiang received the B.Sc. and M.Sc. degreesfrom Xi’an Jiaotong University, Xi’an, China, in1989 and 1992, respectively, and the Ph.D degreefrom Hong Kong University of Science and Tech-nology (HKUST), Hong Kong, in 2000.

He has been working in AT&T labs for four yearsas a senior technical staff member and appointed asAssistant Professor in 2003 and Associated Profes-sors in 2008 at Stevens Institute of Technology. Heis now a Visiting Associate Professor in the HongKong University of Science and Technology. His re-

search focuses on fault detection and activity monitoring using statistical anddata mining methods. His current research interests include data mining and en-terprise intelligence, statistical and data mining methods for quality control andmanagement, as well as logistics management and financial applications.

Dr. Wei Jiang has been awarded a NSF project on business activity monitoringand is a recipient of NSF CAREER award on information quality management.He has published many research papers on quality journals including IIE Trans-actions, Technometrics, Journal of Quality Technology, etc., and is now servingas an associate editor of Journal of Statistical Computation and Simulation. Heis the past chair of the Data Mining Section of the Institute for Operations Re-search and Management Sciences.


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